This research project studies the question of singularity formation in nonlinear dynamical theories of relativistic electromagnetic fields, both at the classical and the quantum level. The classical part is concerned with two main problems: (1) Analysis of the well-known relativistic Vlasov-Maxwell equations, which had been conjectured to be globally well-posed as a Cauchy problem with suitable finite-energy classical data. This project will rigorously analyze a recently-developed scenario for a counterexample exhibiting finite-time collapse for a family of solutions. (2) Analysis of the nonlinear Maxwell-Born-Infeld equations for electromagnetic fields in the absence of point charges. This project will investigate recently-discovered spatially periodic plane wave solutions that exhibit finite-time blow-up to determine whether the corresponding set of Cauchy data is part of a generic bad set. The quantum part of the research project concerns solutions of the Maxwell-Born-Infeld field equations with point defects that represent particles. The charged particles move according to a quantum velocity field obtained from a many body Dirac formalism coupled to generic electromagnetic fields. This project will study whether the Dirac Hamiltonian for the system can become unbounded below. This project addresses fundamental issues in the theory of electromagnetism, which is central to modern science and engineering. The study of singularity formation has long been at the forefront of research in general relativity and in fluid dynamics. Recent discoveries suggest that singularities may also pose a major conceptual challenge in the nonlinear electromagnetic models that have been proposed as candidates for a consistent formulation of an electromagnetic theory without artificial regularizers. The principal investigator recently developed a consistent formulation of electromagnetic theory, incorporating intrinsic spin of particles, that is consistent at both classical and quantum levels. The current project investigates possible singularity formation in this theory. Another part of this work examines finite-time collapse in the relativistic Vlasov-Maxwell model. Such collapse would provide a novel mechanism for the formation of very small celestial bodies whose gravitational self-attraction is too weak to aid in their formation. This project aims to establish that possibility; the results could have a major impact on theories of planetary system formation.
|Effective start/end date||7/1/08 → 6/30/12|
- National Science Foundation (NSF)